Theorem vccl 31171

Description: Closure of the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
vciOLD.1	⊢ 𝐺 = (1st ‘𝑊)
vciOLD.2	⊢ 𝑆 = (2nd ‘𝑊)
vciOLD.3	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
vccl	⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)

Proof of Theorem vccl

Step	Hyp	Ref	Expression
1		vciOLD.1	. . 3 ⊢ 𝐺 = (1st ‘𝑊)
2		vciOLD.2	. . 3 ⊢ 𝑆 = (2nd ‘𝑊)
3		vciOLD.3	. . 3 ⊢ 𝑋 = ran 𝐺
4	1, 2, 3	vcsm 31170	. 2 ⊢ (𝑊 ∈ CVecOLD → 𝑆:(ℂ × 𝑋)⟶𝑋)
5		fovcdm 7592	. 2 ⊢ ((𝑆:(ℂ × 𝑋)⟶𝑋 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)
6	4, 5	syl3an1 1181	1 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   × cxp 5648  ran crn 5651  ⟶wf 6535  ‘cfv 6539  (class class class)co 7421  1^st c1st 8000  2^nd c2nd 8001  ℂcc 11202  CVec_OLDcvc 31166

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-fv 6547  df-ov 7424  df-1st 8002  df-2nd 8003  df-vc 31167

This theorem is used by:  vc0  31182  vcm  31184  nvscl  31234